Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvopab4ndm Structured version   Visualization version   GIF version

Theorem fvopab4ndm 6788
 Description: Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)
Hypothesis
Ref Expression
fvopab4ndm.1 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
Assertion
Ref Expression
fvopab4ndm 𝐵𝐴 → (𝐹𝐵) = ∅)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fvopab4ndm
StepHypRef Expression
1 fvopab4ndm.1 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
21dmeqi 5760 . . . 4 dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
3 dmopabss 5774 . . . 4 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
42, 3eqsstri 3987 . . 3 dom 𝐹𝐴
54sseli 3949 . 2 (𝐵 ∈ dom 𝐹𝐵𝐴)
6 ndmfv 6691 . 2 𝐵 ∈ dom 𝐹 → (𝐹𝐵) = ∅)
75, 6nsyl5 162 1 𝐵𝐴 → (𝐹𝐵) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∅c0 4276  {copab 5114  dom cdm 5542  ‘cfv 6343 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-dm 5552  df-iota 6302  df-fv 6351 This theorem is referenced by:  fvmptndm  6789
 Copyright terms: Public domain W3C validator